(Figs. 9a-f) vary greatly in shape and spatial articulation.
Fig. 9 Design exploration of various funicular funnel shells (a-f) (the tension elements are highlighted in blue) by changing the definition of free or fixed support nodes (marked with blue dots) and the overall magnitude of the horizontal thrusts in each structure
The integration of holes in the shell surface as shown in Fig. 10 demands the topological modification of the form diagram. These openings always form a funicular polygon in the form diagram and are by definition convex; their force equilibrium in the force diagram has a star-shaped topology (Rippmann & Block 2013). The cantilevering ridge edge can be seen as an inverted opening forming a convex boundary.
Fig. 10 Design exploration of various funicular funnel shells with openings
Further, the TNA framework allows controlling the multiple degrees of freedom in statically indeterminate networks. In other words, a statically indeterminate form or force diagram can be geometrically modified while keeping horizontal equilibrium. This means that the length of corresponding elements of the form and force diagram can be modified while guaranteeing their parallel configuration and direction. Consequently, this leads to a local or global increase or decrease of forces since the length of each element in the force diagram represents the horizontal force component of the corresponding element in the structure. In Fig. 11, the force diagram in (b) shows the local attraction of horizontal thrust towards the centre of the structure resulting in a local compression ring and hence a crease in thrust network. It is important to note though that, in contrast to compression shells with fixed supports, the freedom to change the force distribution of structures with continues tension ties is limited due to the geometrically much more strongly constrained force diagram.
Fig. 11 A different force distribution for the same form diagram Γ in (a) and results in a crease in the equilibrium network G of (b)
3.3 Funicular Funnel Rib Vault
In general, the equilibrium stability can effectively be tested with scale models (Van Mele et al., 2012). The presented structural system was tested and verified using a 3D printed, discrete, structural scale model (Fig. 12) to validate the digital results of the form-finding implementation with the physical model. The scale model is made out of discrete - unglued and mechanically not connected - 3D-printed pieces; it is not a structure by itself, but rather an extreme structural model as the pin-jointed, mechanically not connected pieces have a non-rigid (not triangulated) topology. If the solution would not act in pure compression equilibrium and tension along the tension tie, it would immediately become unstable and collapse.
Fig. 12 Structural scale model of a discrete funnel-shaped rib vault
Fig. 13 Equilibrium network G, form diagram Γ and force diagram Γ* of the discrete funnel-shaped rib vault. The colours are directly related to the length of the individual edges in the force diagram and hence visualize the magnitudes of horizontal thrust
As demonstrated by the collapse sequence in Fig. 14, the 3D-printed model is an extreme structural model, as it is made out of discrete, unglued pieces, forming unstable mechanisms, which can only be balanced if the ribs are in perfect axial, compression. The sudden collapse, when cutting the tie, is a demonstration of the structural honesty of the model. As for tree structures or any result of form-finding in general, form-found to act in compression only for a defined, dominant loading case (Lachauer &
Fig. 14 Collapse of the 3D-printed funnel-shaped rib vault by cutting the continuous tension tie
Block, 2012), this type of structures need to be realized with stiff nodes in order to resist live, non-funicular loading cases, which can be defined and dimensioned through the analysis of a materialized, form-found shape.
4 Conclusion and Further Research
This paper discussed how funnel geometries could be made structurally efficient, as a combination of a three-dimensional equivalent of funicular half-arches balanced by tension ties. It showed how Thrust Network Analysis could be extended to incorporate tension elements, using directed elements in the form and force diagram, to create continuous tension rings. These concepts have been implemented by extending RhinoVAULT to these new boundary conditions. A simple design exploration showed the variety of possible shapes using only simple modification strategies. Lastly, a structural model was developed to validate the equilibrium solutions generated with the approach. This model furthermore hints at the attractive possibilities of filigree ribbed funnel structures, reminiscent of Schlaich’s beautiful tree structures.
Although the current, direct implementation allows an intuitive exploration of funicular funnel shells, as clear from the exploration in the result section, the following objectives represent routes for further research:
Understand the actual dependencies and constraints of the form and force diagram for this new type of boundary condition better to fully explore the possibilities of the funicular funnel shell typology;
Include feedback, or immediately constraints, on the global moment equilibrium during the form-finding process: Shells whose centres of gravity fall outside of the convex hull of the supports, can of course not stand as a combined pure compression shell and tension rings; and
Formulate the form-finding as a best-fit (to a target surface/geometry) optimization problem, as in Block & Lachauer (2011) or Panozzo et al. (2013) for compression-only shells.
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Panozzo,